A Graph Showing a Negative Relationship

A Graph Showing a Negative Relationship

As an example of a graph of a negative relationship, let us look at the impact of the cancellation of games by the National Basketball Association during the 1998–1999 labor dispute on the earnings of one player: Shaquille O’Neal. During the 1998–1999 season, O’Neal was the center for the Los Angeles Lakers.

O’Neal’s salary with the Lakers in 1998–1999 would have been about $17,220,000 had the 82 scheduled games of the regular season been played. But a contract dispute between owners and players resulted in the cancellation of 32 games. Mr. O’Neal’s salary worked out to roughly $210,000 per game, so the labor dispute cost him well over $6 million. Presumably, he was able to eke out a living on his lower income, but the cancellation of games cost him a great deal.

We show the relationship between the number of games canceled and O’Neal’s 1998–1999 basketball earnings graphically in Figure 21.5 "Canceling Games and Reducing Shaquille O’Neal’s Earnings". Canceling games reduced his earnings, so the number of games canceled is the independent variable and goes on the horizontal axis. O’Neal’s earnings are the dependent variable and go on the vertical axis. The graph assumes that his earnings would have been $17,220,000 had no games been canceled (point A, the vertical intercept). Assuming that his earnings fell by $210,000 per game canceled, his earnings for the season were reduced to $10,500,000 by the cancellation of 32 games (point B). We can draw a line between these two points to show the relationship between games canceled and O’Neal’s 1998–1999 earnings from basketball. In this graph, we have inserted a break in the vertical axis near the origin. This allows us to expand the scale of the axis over the range from $10,000,000 to $18,000,000. It also prevents a large blank space between the origin and an income of $10,500,000—there are no values below this amount.

What is the slope of the curve in Figure 21.5 "Canceling Games and Reducing Shaquille O’Neal’s Earnings"? We have data for two points, A and B. At A, O’Neal’s basketball salary would have been $17,220,000. At B, it is $10,500,000. The vertical change between points A and B equals -$6,720,000. The change in the horizontal axis is from zero games canceled at A to 32 games canceled at B. The slope is thus

Notice that this time the slope is negative, hence the downward-sloping curve. As we travel down and to the right along the curve, the number of games canceled rises and O’Neal’s salary falls. In this case, the slope tells us the rate at which O’Neal lost income as games were canceled.

The slope of O’Neal’s salary curve is also constant. That means there was a linear relationship between games canceled and his 1998–1999 basketball earnings.

Shifting a Curve

Consider, for example, the ski club that sponsors bus trips to the ski area. The graph we drew in Figure 21.2 "Plotting a Graph" shows the relationship between club revenues from a particular trip and the number of passengers on that trip, assuming that all other variables that might affect club revenues are unchanged. Let us change one. Suppose the school’s student government increases the payment it makes to the club to $400 for each day the trip is available. The payment was $200 when we drew the original graph. Panel (a) of Figure 21.6 "Shifting a Curve: An Increase in Revenues"shows how the increase in the payment affects the table we had in Figure 21.1 "Ski Club Revenues"; Panel (b) shows how the curve shifts. Each of the new observations in the table has been labeled with a prime: A′, B′, etc. The curve R₁ shifts upward by $200 as a result of the increased payment. A shift in a curve implies new values of one variable at each value of the other variable. The new curve is labeled R₂. With 10 passengers, for example, the club’s revenue was $300 at point B on R₁. With the increased payment from the student government, its revenue with 10 passengers rises to $500 at point B′ on R₂. We have a shift in the curve.

It is important to distinguish between shifts in curves and movements along curves. Amovement along a curve is a change from one point on the curve to another that occurs when the dependent variable changes in response to a change in the independent variable. If, for example, the student government is paying the club $400 each day it makes the ski bus available and 20 passengers ride the bus, the club is operating at point C′ on R₂. If the number of passengers increases to 30, the club will be at point D′ on the curve. This is a movement along a curve; the curve itself does not shift.

Now suppose that, instead of increasing its payment, the student government eliminates its payments to the ski club for bus trips. The club’s only revenue from a trip now comes from its $10/passenger charge. We have again changed one of the variables we were holding unchanged, so we get another shift in our revenue curve. The table in Panel (a) of Figure 21.7 "Shifting a Curve: A Reduction in Revenues" shows how the reduction in the student government’s payment affects club revenues. The new values are shown as combinations A″ through E″ on the new curve, R₃, in Panel (b). Once again we have a shift in a curve, this time from R₁ to R₃.

Figure 21.7 Shifting a Curve: A Reduction in Revenues



The table in Panel (a) shows the impact on ski club revenues of an elimination of support from the student government for ski bus trips. The club’s only revenue now comes from the $10 it charges to each passenger. The new combinations are shown as A″ – E″. In Panel (b) we see that the original curve relating club revenue to the number of passengers has shifted down.

The shifts in Figure 21.6 "Shifting a Curve: An Increase in Revenues" and Figure 21.7 "Shifting a Curve: A Reduction in Revenues" left the slopes of the revenue curves unchanged. That is because the slope in all these cases equals the price per ticket, and the ticket price remains unchanged. Next, we shall see how the slope of a curve changes when we rotate it about a single point.